Visualizing Fractals Using 3D Printing Mathematical Ted Spiro Divye Bhagwani Harsha Podapati Supervisor: Dr. Olga Lukina Computing Laboratory University of Illinois at Chicago

Summary Some of What We Made Box Counting Dimension Challenges

The goal of this research project was to create 3D models of var- As the name suggests, we can count how many boxes it takes One of the biggest challenges we overcame was finding the best ious self-similar fractals to better understand their mathematical to cover the object in order to calculate its dimension. The di- software to model these designs. We needed a software that could properties by having a physical model to manipulate and analyze. mension is calculated by zooming in (that is, making the boxes both do the operations we required as well as generate 3D ren- We did this by researching how the fractals are created, and then smaller) and re-computing the number of boxes. derings. Our research first started on Wolfram Mathematica. But, creating a program to render a model of the design. After a digital due to the limited nature of the Online version, we shifted to us- model was created, we exported the rendering to the 3D model- So, we get the following equation for the box dimension: ing OpenSCAD. This change put us away from using higher level ing software for the printer to convert the model into a 3D printing functions, but gave us more customizability to our programs. log(N (S)) log(N (S)) blueprint. Then we would print the model and hone the settings Box(S) = lim inf δ = lim sup δ δ→0 1 δ→0 1 until we ended up with a perfect print. log(δ ) log(δ ) Another challenge of this project was to make models that didn’t turn out like this... where S is the given set or fractal design, δ is the length of the sides of the boxes, and N is the number of boxes. Equipment & Software δ In general, the values of liminf and limsup in the formula above I Ultimaker 2 3D Printer with PLA(Polylactic Acid) Filament may be non-equal. In that case, the box dimension of the fractal I Ultimaker Cura (3D Printing Software) is not defined, but we can still talk about the lower box counting I OpenSCAD dimension, and the upper box counting dimension of S. This required us to understand all the mechanics of 3D print- I Wolfram Mathematica Now that we have the formula, let’s compute the box counting ing. We learned that different filament colors printed differently: dimension for the Menger Sponge. Let darker colors were more oily than lighter colors. Learning what 1 1 the settings should be for each print and filament was crucial to What is a Fractal ≤ δ ≤ . get the successful prints that we made. 3n 3n−1 Fractals are patterns or objects that are self-similar across differ- Lastly, we learned that many fractals cannot be successfully 3D When we create the Menger Sponge by substitution, we divide a ent scales. They are created by repeating a process infinitely many printed due to physical constraints. The first being that fractals cube into 27 smaller cubes and remove the middle 7 cubes. times. are objects that have processes that have taken place an infinite amount of times. Since we can never reach infinity, we can never Since the main objective of this project is 3D printing fractal ob- have a fully realized fractal. Also, many fractal objects were too jects, we printed various well-known fractals, as well as less com- complex to render due to computational limitations. mon types. We also managed to print some 2 dimensional fractal When we do this, we are left with 20 cubes. Repeating this pro- objects by giving them a very small height. cedure n times, we have: Because of this, there were many fractals that we created that were not able to be physically realized. n−1 n 20 ≤ Nδ (S) ≤ 20 How To Create Fractals After plugging these into the equation and doing some tedious algebra, we find the following: Fractals are created by recursion. A recursion is a procedure Outcome which invokes itself. log20 Box(S) = ≈ 2.72683 Even through our challenges, we had a very successful semester, log3 creating numerous fractal objects using our 3D printer. We also Substitution: This method is a form of recursion, where part of learned a lot about creating fractals and about their many interest- the object is removed or replaced according to a substitution rule. This means that the dimension of the fractal is less than 3 but ing mathematical properties. For example, in the second picture a cube is divided into 27 cubes, greater than 2 when iterations tend towards infinity! then the middle 7 cubes are removed. The same substitution rule Lastly, we would like to thank Dr. Olga Lukina for her guidance is then applied to each of the remaining cubes recursively. Doing this same process for the Sierpinski Pyramid yields a Box and teaching throughout the semester. Without her knowledge Counting Dimension of 2. This means that, mathematically, the and help, this project would not have been a success. Iteration: This method involves fractals constructed by the union fractal has no volume! of several copies of themselves. Each copy is transformed by a function. The method is also referred to as the Iterated Function References System. Hausdorff Dimension Christopher J. Bishop and Yuval Peres. Fractals in Proba- The Hausdorff dimension is very similar to the box counting di- bility and Analysis. Cambridge Studies in Advanced Mathe- List of Printed Objects mension. However, there is one key difference: when computing matics. Cambridge University Press, 2016. DOI: 10 . 1017 / the Hausdorff dimension, we cover the fractal S by boxes with 9781316460238. st I Sierpinski Triangle and Sierpinski Pyramid (1 picture) edges of length at most δ. When computing the Box Dimension, David Mumford, Caroline Series, and David Wright. Indra’s nd The Math Behind the Prints I Menger Carpet and Menger Sponge (2 picture) we cover S by boxes with edges of length precisely δ. So for Pearls: The Vision of Felix Klein. Cambridge University Press, rd some sets, the Hausdorff and the box counting dimensions are not I Fractal Trees (3 picture) A mathematical fractal is constructed by repeating a recursive 2002. DOI: 10.1017/CBO9781107050051. th equal. I Dodecahedron Fractal (4 picture) procedure infinitely many times. Such fractals often have frac- Wikibooks. OpenSCAD User Manual/The OpenSCAD Lan- Cube Fractal (5th picture) I tional dimension. For example, the set K = {0}S{1, 1, 1,···} has box dimension guage — Wikibooks, The Free Textbook Project. [Online; Koch Snowflake 2 3 I 1/2 and Hausdorff dimension 0 (Example 1.8 in the book by accessed 21-November-2018]. 2017. URL: https : / / en . Although it is physically impossible to print a fractal constructed Bishop and Peres). wikibooks.org/w/index.php?title=OpenSCAD_User_ by an infinite number of iterations, it is worth exploring the math- Manual/The_OpenSCAD_Language&oldid=3289314. ematical concept of fractional dimension. http://mcl.math.uic.edu/